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Mass Spectrometry

Field: Spectroscopy

An analytical technique that measures the mass-to-charge ratio of ions.

Sequence of Expressions

Theorem

Ionization

Define the ionization process I:M→M++e−\mathcal{I}: M \rightarrow M^+ + e^-. If the process is modeled by electron impact ionization (EI) with incident electron energy EincE_{inc}, the resulting ion M+M^+ is characterized by the conservation of energy and momentum, such that the internal energy EintE_{int} of the resulting ion is given by:\n⟨Eint⟩=Einc−Ebind−Ekin\langle E_{int} \rangle = E_{inc} - E_{bind} - E_{kin}
Let mm be the measured mass of an ion (in atomic mass units, uu) and zz be the net charge of the ion (an integer). The mass-to-charge ratio, RR, is defined as:\nR=mzR = \frac{m}{z}
Consider a parent ion M+M^+ with mass mMm_M. Fragmentation is modeled as a decomposition into a set of daughter ions {Fi+}\{F_i^+\} and neutral fragments NjN_j. The mass balance requires that the sum of the masses of the products equals the mass of the parent ion, subject to the conservation of charge:\nM+→∑i=1kFi++∑j=1lNjM^+ \rightarrow \sum_{i=1}^{k} F_i^+ + \sum_{j=1}^{l} N_j \nMass Conservation: mM=∑i=1kmFi+∑j=1lmNj\text{Mass Conservation: } m_M = \sum_{i=1}^{k} m_{F_i} + \sum_{j=1}^{l} m_{N_j}
The operational requirement for a high vacuum system is maintaining a pressure PP significantly below atmospheric pressure, typically P<10−5P < 10^{-5} Torr. This condition ensures that the mean free path λ\lambda of the ions is much greater than the dimensions of the instrument, such that the collision frequency νcoll\nu_{coll} with residual gas molecules is minimized:\nλ=kBTPσ≫L\lambda = \frac{k_B T}{P \sigma} \gg L \nwhere kBk_B is the Boltzmann constant, TT is the temperature, PP is the pressure, σ\sigma is the collision cross-section, and LL is the characteristic length scale.
Theorem

Resolution

The resolution RR of a mass spectrometer is defined as the ratio of the measured mass mm to the full width at half maximum (FWHM) of the peak function P(m)\mathcal{P}(m) at that mass:\nR=mΔm1/2R = \frac{m}{\Delta m_{1/2}} \nFor a Gaussian peak profile, P(m)=Aexp⁡(−(m−m0)22σ2)\mathcal{P}(m) = A \exp\left(-\frac{(m - m_0)^2}{2\sigma^2}\right), the resolution is mathematically related to the standard deviation σ\sigma by:\nR=22ln⁡2⋅mσR = \frac{2\sqrt{2\ln 2} \cdot m}{\sigma}
Let m1m_1 and m2m_2 be the masses of two isotopes, and qq be the charge. The separation factor α\alpha for the mass-to-charge ratio m/zm/z is defined by the ratio of the measured ion intensities II: \nα=I(m1/q)I(m2/q)≈m1m2\alpha = \frac{I(m_1/q)}{I(m_2/q)} \approx \frac{m_1}{m_2} \nFor high-resolution separation, the differential abundance ΔA\Delta A of an isotope mim_i relative to a reference isotope mrefm_{ref} is modeled by the ratio of their natural abundances AiA_i: \nAiAref=NiNref≈mimref⋅e−(mi−mref)22σ2\frac{A_i}{A_{ref}} = \frac{N_i}{N_{ref}} \approx \frac{m_i}{m_{ref}} \cdot e^{-\frac{(m_i - m_{ref})^2}{2\sigma^2}} \nwhere σ\sigma is the mass resolution parameter.
Define the ionization process I\mathcal{I} for a neutral molecule MM subjected to an electron beam with energy Ee−E_{e^-}. The resulting ion M+∙M^{+\bullet} is formed via collision: \nM+e−(Ee−)→M+∙+2e−+EkinM + e^- (E_{e^-}) \rightarrow M^{+\bullet} + 2e^- + E_{kin} \nAssuming the ionization potential IPIP is the minimum energy required, the excess kinetic energy EexcessE_{excess} of the resulting ion is given by the energy conservation principle: \nEexcess=Ee−−IPE_{excess} = E_{e^-} - IP \nIf the collision is modeled by a scattering cross-section σcoll\sigma_{coll}, the ion current IionI_{ion} is proportional to the electron flux Φe−\Phi_{e^-} and the cross-section: \nIion∝Φe−⋅σcoll(Ee−)I_{ion} \propto \Phi_{e^-} \cdot \sigma_{coll}(E_{e^-})
Define the potential applied to the quadrupole rods by the voltages UU (DC) and VV (RF) as the potential function Φ\Phi: \nΦ(x,y,z)=UR2(x2−y2)+VR2(x2−y2)cos⁡(ωt)\Phi(x, y, z) = \frac{U}{R^2} (x^2 - y^2) + \frac{V}{R^2} (x^2 - y^2) \cos(\omega t) \nFor an ion with mass-to-charge ratio m/zm/z to maintain a stable trajectory within the quadrupole field, its motion must satisfy the Mathieu equation for both xx and yy coordinates: \nd2dξ2(ξxξy)+(aκ−κa)(ξxξy)=0\frac{d^2}{d\xi^2} \begin{pmatrix} \xi_x \\ \xi_y \end{pmatrix} + \begin{pmatrix} a & \kappa \\ -\kappa & a \end{pmatrix} \begin{pmatrix} \xi_x \\ \xi_y \end{pmatrix} = 0 \nwhere ξ=ωt\xi = \omega t, and the parameters aa and κ\kappa are functions of the applied voltages U,VU, V and the ion's m/zm/z ratio.
Let Iion(t)I_{ion}(t) be the instantaneous ion current (Amperes) detected at time tt. The detector converts this current into a measurable voltage signal Vout(t)V_{out}(t) via a transimpedance amplifier (TIA) with a gain GG: \nVout(t)=−Iion(t)⋅RfeedbackV_{out}(t) = -I_{ion}(t) \cdot R_{feedback} \nFor quantitative analysis, the measured signal SS is the integral of the ion current over the acquisition time Δt\Delta t: \nS=∫t0t0+ΔtIion(t)dtS = \int_{t_0}^{t_0 + \Delta t} I_{ion}(t) dt \nIf the detector response is modeled by a Gaussian function R(t)\mathcal{R}(t) centered at tpeakt_{peak}, the peak signal amplitude SpeakS_{peak} is proportional to the ion abundance NN: \nSpeak=k⋅N⋅e−(t−tpeak)22σdet2S_{peak} = k \cdot N \cdot e^{-\frac{(t - t_{peak})^2}{2\sigma_{det}^2}} \nwhere kk is the detector sensitivity constant.
Define the van 't Hoff factor ii for the dissociation of a solute in a solvent. If the solute dissociates into nn ions, the theoretical factor is itheory=ni_{theory} = n. The observed van 't Hoff factor iobsi_{obs} relates the measured concentration of ions [Cion][C_{ion}] to the initial concentration [C0][C_0]: \niobs=[Cion][C0]i_{obs} = \frac{[C_{ion}]}{[C_0]} \nIn the context of mass spectrometry, the observed intensity IobsI_{obs} of a molecular ion peak relative to the theoretical intensity ItheoryI_{theory} is related by the factor ii: \nIobs=i⋅ItheoryI_{obs} = i \cdot I_{theory} \nwhere ii accounts for non-ideal solution behavior and incomplete ionization.